3.7: Pythagorean Theorem, Part 2: Applications & Triples

Learning Objectives

Identify common Pythagorean triples.

Using Pythagorean Triples

Review the example problems from the previous lesson.

This is the diagram from Example 1:

In Example 1, the sides of the triangle are 3, 4, and 5. This combination of numbers is referred to as a Pythagorean triple. A Pythagorean triple is three integers (whole numbers with no decimal or fraction part) that make the Pythagorean Theorem true.

A Pythagorean triple is a group of three _____________________________ that satisfy the Pythagorean Theorem.

Throughout this chapter, you will learn other Pythagorean triples as well.

This is the diagram from Example 2:

Using the Pythagorean Theorem equation , and letting and , we calculated that inches.

The triangle in Example 2 is proportional to the same ratio of 3 : 4 : 5. If you divide the lengths of the triangle (6, 8, and 10) by 2, you find the same proportion — 3 : 4 : 5 (because , and ).

Whenever you find a Pythagorean triple, you can apply these ratios with greater factors as well.

Finally, look at the side lengths of the triangle in Example 3:

The two legs are 5 cm and 12 cm and the length of the missing side (the hypotenuse) is 13 cm. The side lengths make a ratio of 5 : 12 : 13. This, too, is a Pythagorean triple. You can infer that this ratio, multiplied by greater factors, will also yield numbers that satisfy the Pythagorean Theorem.

There are infinitely many Pythagorean triples, but a few of the most common ones and their multiples are in the chart below:

Pythagorean triple

3 – 4 – 5

6 – 8 – 10

9 – 12 – 15

12 – 16 – 20

5 – 12 – 13

10 – 24 – 26

15 – 36 – 39

20 – 48 – 52

7 – 24 – 25

14 – 48 – 50

21 – 72 – 75

28 – 96 – 100

8 – 15 – 17

16 – 30 – 34

24 – 45 – 51

32 – 60 – 68

Reading Check:

1. What is a Pythagorean triple?

2. Which of the following is NOT a Pythagorean triple? Show your work.

a. 15 – 36 – 39

b. 15 – 20 – 25

c. 16 – 30 – 35

d. 25 – 60 – 65

3. Give 2 examples of Pythagorean triples that are NOT in the chart above.